#497937

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

510 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405133);