#7186714

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1717 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)(8,16)(11,17)(15,20)(18,19)(21,22,28,33,27,23,26,30)(24,29,35,25,32,31,34,36);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,22)(24,29,32,31)(25,35,36,34)(26,30,28,33);(1,2,3,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,29)(23,31)(25,33)(26,34)(27,32)(28,35)(30,36);(7,14)(10,13)(15,20)(18,19)(21,25,28,31,27,36,26,29)(22,24,33,35,23,32,30,34);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,23)(24,32)(30,33)(34,35)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=g,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=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, 7093910);