#7592219

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	#44869	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1746 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,22,25)(19,26,28,31)(20,21,24,29)(23,30,32,27);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18,22,25)(19,26,28,31)(20,21,24,29)(23,30,32,27);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19,21,27,22,28,29,30)(18,23,20,26,25,32,24,31);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20,21,25,22,24,29,18)(19,26,27,32,28,31,30,23);(2,6)(7,8)(10,14)(13,15)(17,21,22,29)(18,24,25,20)(19,27,28,30)(23,31,32,26)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=fg,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7499415);