#476962

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(5,6)(7,10)(8,11)(9,12)(13,14,18,23)(15,20,28,37)(16,31,17,21)(19,24,35,25)(22,33,40,32)(26,41,43,36)(27,44,34,39)(29,38,30,42);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,27)(17,34)(18,28)(19,30)(21,39)(22,41)(23,37)(24,42)(25,38)(26,32)(29,35)(31,44)(33,43)(36,40);(2,3)(5,7)(6,10)(8,12)(9,11)(13,16,18,17)(14,21,23,31)(15,26,28,43)(19,33,35,32)(20,36,37,41)(22,25,40,24)(27,29,34,30)(38,44,42,39);(2,3)(7,12)(9,10)(13,17,18,16)(14,22,23,40)(15,27,28,34)(19,33,35,32)(20,36,37,41)(21,25,31,24)(26,29,43,30)(38,39,42,44)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi,h^d=hi

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, 384158);