#7622868

order	512
nilpotency class	4
isoclinism	isoclinic class of #33551
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×D₄	C₂⁵
socle	C₂	C₂²×B(4,𝔽₂)
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|e^2=i,b^a=bf,c^a=cg,h^a=hi,d^b=di,g^b=ghi,f^c=fh,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218460, #72231746, #72233471, …
maximal normal subgroup of	#72218460, #72231746, #72234211, …
maximal quotient of	#72234768, #72234794, C₂×#7622868, …
automorphism group of	#1320, #1321, #1327

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530064);