#7532106

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	#45353	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1871 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,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,e^d=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281760, #72282274, #72282729
maximal normal subgroup of	#72281760, #72282274, #72282729

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7439302);