#349028

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

902 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=hi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^b=di,f^c=fh,g^c=gi,f^d=fi,g^d=gi

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, 256224);