#297342

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
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	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1057 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fh,d^2=ei,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204538);