#770019

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1998 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)(6,10)(7,12)(13,14,17,16)(15,19,20,18)(21,22)(23,29)(24,31)(25,33)(26,27)(28,35)(30,32)(34,36);(13,14,17,16)(15,19,20,18)(22,27)(24,29)(28,33)(32,36);(5,6)(7,11)(8,12)(9,10)(13,15,17,20)(14,18,16,19)(21,23)(22,24)(25,34)(26,31)(27,29)(28,36)(30,35)(32,33);(1,2,3,4)(5,7,9,12)(6,11,10,8)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,34)(35,36);(5,8,9,11)(6,12,10,7)(13,16,17,14)(15,19,20,18)(21,25,26,35)(22,28,27,33)(23,30,31,34)(24,32,29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 677215);