#6545555

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1609 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)(3,8)(4,6)(5,7)(17,18,23,24,22,26,19,27)(20,29,28,21,25,31,30,32);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,26,27)(20,30,25,28)(21,31,32,29);(10,13)(14,16)(17,20,22,25)(18,21,26,32)(19,28,23,30)(24,29,27,31);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,21)(18,25)(19,29)(20,26)(22,32)(23,31)(24,30)(27,28);(11,15)(14,16)(18,26)(21,32)(24,27)(29,31)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452751);