#7613456

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281313, #72281624
maximal normal subgroup of	#72281313, #72281624

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520652);