#7544224

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#25087	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1623 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,24)(21,23)(22,30)(26,27)(28,31)(29,32);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,29)(21,31)(22,27)(23,28)(24,32)(26,30);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,22)(19,26)(21,24)(23,30)(25,32)(27,31)(28,29);(10,14)(13,15)(17,21,23,18)(19,26,28,29)(20,22,30,24)(25,32,31,27);(2,5)(6,8)(10,14)(13,15)(17,22,23,24)(18,20,21,30)(19,27,28,32)(25,29,31,26)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fgi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308745, #72308782
maximal normal subgroup of	#72308745, #72308782

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7451420);