#7061419

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#23580	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1497 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,d^2=f,e^2=f,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=dhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308825, #72308985, #72309031
maximal normal subgroup of	#72308825, #72308985, #72309031

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6968615);