#496137

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₁₆	C₂³
commutator, socle, prime radical	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

550 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)(13,14,18,23,19,24,33,39,17,22,30,38,32,16,21,31)(15,25,28,41,29,27,40,43,26,20,34,36,42,37,35,44);(5,7,9,8)(6,10,11,12)(13,15,17,26)(14,20,22,25)(16,27,24,37)(18,28,30,34)(19,29,32,42)(21,35,33,40)(23,36,38,41)(31,43,39,44);(1,2,3,4)(15,26)(20,25)(27,37)(28,34)(29,42)(35,40)(36,41)(43,44);(5,8,9,7)(6,10,11,12)(13,16,30,39,19,14,21,38,17,24,18,31,32,22,33,23)(15,27,34,44,29,20,35,41,26,37,28,43,42,25,40,36)

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403333);