#498617

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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	C₂²×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

576 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=efhi,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405813);