#485755

order	512
nilpotency class	4
isoclinism	isoclinic class of #485649
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	#47664	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 799 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)(19,27)(20,24)(21,34)(22,26)(23,30)(25,36)(28,42)(29,43)(31,45)(32,47)(33,41)(35,37)(38,44)(39,46)(40,48);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(19,28)(23,37)(25,34)(29,31)(30,35)(38,39)(40,47)(44,46);(17,19,22,28)(18,23,26,37)(20,31,33,29)(21,35,36,30)(24,39,41,38)(25,42,34,27)(32,44,48,46)(40,43,47,45);(3,10)(6,15)(9,16)(11,13)(17,20,22,33)(18,24,26,41)(19,29,28,31)(21,32,36,48)(23,38,37,39)(25,40,34,47)(27,43,42,45)(30,44,35,46)

Polycyclic presentation:

2^9|a^2=g,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,d^c=di,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	#72285896
maximal normal subgroup of	#72285896

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392951);