#497918

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂²×D₄	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

628 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)(6,9)(10,12)(13,14)(15,19)(16,27)(17,32)(18,23)(20,31)(21,25)(22,26)(24,39)(28,41)(29,38)(30,42)(33,35)(34,40)(36,43)(37,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,23,22)(16,28,30,24)(17,25,33,29)(20,36,37,34)(21,35,38,32)(27,41,42,39)(31,43,44,40);(5,7,8,11)(6,10,9,12)(13,16,29,43,15,24,17,31,18,30,25,40,26,28,33,44)(14,20,35,39,22,36,21,27,23,37,32,41,19,34,38,42);(6,9)(10,12)(16,30)(20,37)(24,28)(27,42)(31,44)(34,36)(39,41)(40,43)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=efhi,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,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, 405114);