#297512

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#9914	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1269 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dhi,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204708);