#485742

order	512
nilpotency class	4
isoclinism	isoclinic class of #485657
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

780 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,9,10,16)(4,12,14,7)(6,13,15,11)(17,18)(19,24)(20,32)(21,26)(22,36)(23,28)(25,34)(27,38)(29,40)(30,46)(31,42)(33,48)(35,44)(37,41)(39,45)(43,47);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,19,23,31)(18,24,28,42)(20,33,21,29)(22,37,38,30)(25,43,44,39)(26,40,32,48)(27,41,36,46)(34,45,35,47);(17,20,23,21)(18,25,28,44)(19,29,31,33)(22,35,38,34)(24,39,42,43)(26,36,32,27)(30,47,37,45)(40,41,48,46);(3,10)(6,15)(9,16)(11,13)(17,21,23,20)(18,26,28,32)(19,29,31,33)(22,35,38,34)(24,40,42,48)(25,36,44,27)(30,45,37,47)(39,41,43,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228029, #72228033, #72228035
maximal normal subgroup of	#72228029, #72228033, #72228035

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392938);