#7166648

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993915
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1804 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:

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

Polycyclic presentation:

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

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, 7073844);