#498199

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	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

594 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:

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

Polycyclic presentation:

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

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, 405395);