#495520

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

876 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402716);