#348996

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

912 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)(5,6,9,12)(7,13,15,20)(8,16,18,11)(10,17,19,14)(21,22,24,27)(23,31,32,36)(25,28,26,30)(29,33,35,34);(1,2,3,4)(22,27)(23,32)(28,30)(33,34);(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13)(22,28)(25,26)(27,30)(29,36)(31,35)(33,34);(21,23,24,32)(22,29,27,35)(25,34,26,33)(28,36,30,31)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,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, 256192);