#493327

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

570 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=e,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=de,c^b=cei,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400523);