#451117

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 ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1212 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)(6,9)(10,11)(13,14,16,18)(15,17,19,20)(21,22)(23,27)(24,32)(25,29)(26,30)(28,31)(33,35)(34,36);(5,6,8,9)(7,11,12,10)(13,15)(14,17)(16,19)(18,20)(21,23,26,25)(22,27,30,29)(24,31,33,34)(28,35,36,32);(15,19)(17,20)(21,24,26,33)(22,28,30,36)(23,31,25,34)(27,35,29,32);(2,4)(5,7,8,12)(6,10,9,11)(21,23,26,25)(22,29,30,27)(24,31,33,34)(28,32,36,35)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ei,c^2=i,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=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, 358313);