#7498553

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309020
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	#32681	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=i,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7405749);