#7198625

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(21,22,25,29,26,30,34,27)(23,28,32,33,31,36,35,24);(5,7,10,6)(8,11,15,13)(9,12,17,14)(16,19,20,18)(22,27)(24,28)(25,34)(29,30)(32,35)(33,36);(1,2,3,4)(5,8,10,15)(6,11,7,13)(9,16,17,20)(12,18,14,19)(21,23,26,31)(22,28,30,36)(24,29,33,27)(25,32,34,35);(8,15)(11,13)(16,20)(18,19)(21,24,25,28,26,33,34,36)(22,23,29,32,30,31,27,35);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(23,31)(24,33)(28,36)(32,35)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=ghi,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7105821);