#495378

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^c=di

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, 402574);