#363326

order	512
nilpotency class	3
isoclinism	isoclinic class of #23918
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1286 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=i,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270522);