#496139

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

532 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=eg,g^2=h,h^2=i,b^a=be,c^a=cf,c^b=ci,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, 403335);