#7622760

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3483	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1363 conjugacy classes of subgroups; only the 1018 normal subgroups are shown.

normal subgroups only

This group has 1363 conjugacy classes of subgroups — too many to draw. Only its 1018 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,e^2=g,g^2=h,h^2=i,b^a=bf,c^a=ci,d^a=dfi,c^b=cfi,d^b=di,d^c=di

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529956);