#497935

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fh,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, 405131);