#444741

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1024 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)(7,9)(8,10)(11,12)(13,14,16,18)(15,17,19,20)(22,26)(24,32)(27,33)(28,35)(29,36)(30,34);(5,7,8,11)(6,9,10,12)(15,19)(17,20)(21,22,25,27)(23,29,31,35)(24,33,32,26)(28,34,36,30);(6,10)(9,12)(13,15,16,19)(14,17,18,20)(22,27)(26,33)(28,36)(29,35);(2,4)(21,23,25,31)(22,28,27,36)(24,34,32,30)(26,35,33,29)

Polycyclic presentation:

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

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, 351937);