#7610258

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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	#59097	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1674 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)(11,13)(12,14)(15,16)(17,18,22,25)(19,28,29,24)(20,26,30,32)(21,31,23,27);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,21)(22,29)(23,30)(25,28)(26,27)(31,32);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(18,25)(19,29)(20,30)(27,31);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,20,21,19,22,30,23,29)(18,26,31,28,25,32,27,24);(2,6)(7,8)(17,21,22,23)(18,27,25,31)(19,20,29,30)(24,26,28,32)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=fh,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7517454);