#495376

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
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

894 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)(13,14)(15,26)(16,24)(17,23)(18,34)(19,35)(20,25)(21,29)(22,31)(27,37)(28,36)(30,42)(32,40)(33,41)(38,44)(39,43);(2,4)(6,9)(10,11)(13,15,16,27,19,28,31,43,20,30,32,44,17,33,18,29)(14,21,34,41,23,38,40,42,25,39,22,36,35,37,24,26);(5,7,8,12)(6,10,9,11)(13,16,19,31,20,32,17,18)(14,22,23,24,25,34,35,40)(15,27,28,43,30,44,33,29)(21,36,38,26,39,41,37,42);(13,17,20,19)(14,23,25,35)(15,28,30,33)(16,18,32,31)(21,37,39,38)(22,24,34,40)(26,36,42,41)(27,43,44,29)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=i,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^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, 402572);