#485704

order	512
nilpotency class	4
isoclinism	isoclinic class of #485617
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×C₂²⋊C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 810 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)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,43)(32,44)(33,48)(34,40)(35,41)(36,42)(37,45)(38,47)(39,46);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,19,23,32)(18,24,28,38)(20,33,21,30)(22,31,36,47)(25,39,26,37)(27,29,42,44)(34,46,35,45)(40,48,41,43);(17,20,23,21)(18,25,28,26)(19,30,32,33)(22,35,36,34)(24,37,38,39)(27,40,42,41)(29,43,44,48)(31,46,47,45);(3,10)(6,15)(9,16)(11,13)(17,21,23,20)(18,26,28,25)(19,30,32,33)(22,35,36,34)(24,37,38,39)(27,41,42,40)(29,43,44,48)(31,45,47,46)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392900);