#6653995

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37278	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6561191);