#7025036

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1580 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6932232);