#366809

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1372 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274005);