#93065

order	512
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#331
commutator	C₄×C₈	C₄²
Frattini, cosocle	C₂×C₈²	C₂²
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,e^2=fghi,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261);