#497955

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

502 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)(6,10)(11,12)(13,14)(15,19)(16,27)(17,31)(18,23)(20,29)(21,25)(22,26)(24,38)(28,37)(30,42)(32,34)(33,39)(35,43)(36,44)(40,41);(5,6,9,10)(7,11,8,12)(13,15,18,26)(14,19,23,22)(16,24,30,40)(17,25,32,28)(20,33,36,35)(21,34,37,31)(27,38,42,41)(29,39,44,43);(5,7,9,8)(6,11,10,12)(13,16,28,43,15,24,17,29,18,30,25,39,26,40,32,44)(14,20,34,38,22,35,21,27,23,36,31,41,19,33,37,42);(5,8,9,7)(6,11,10,12)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=efhi,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, 405151);