#6856465

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#32913	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1581 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285273, #72285539
maximal normal subgroup of	#72285273, #72285539

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6763661);