#460317

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367513);