#496055

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=gi,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,c^b=ci,d^b=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, 403251);