#503542

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

809 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=efh,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410738);