#385991

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1297 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)(7,9)(8,10)(11,12)(13,14)(15,24)(16,20)(17,29)(18,23)(19,30)(21,26)(22,34)(25,41)(27,28)(31,44)(32,40)(33,39)(35,37)(36,42)(38,43);(5,7,9,6)(8,11,12,10)(13,15,18,26)(14,21,23,24)(16,28,30,37)(17,31,32,25)(19,27,20,35)(22,38,39,36)(29,41,40,44)(33,43,34,42);(5,8,9,12)(6,10,7,11)(13,16,19,14,18,30,20,23)(15,24,35,37,26,21,27,28)(17,29,34,22,32,40,33,39)(25,38,42,41,31,36,43,44);(6,7)(10,11)(13,17,18,32)(14,22,23,39)(15,25,26,31)(16,29,30,40)(19,34,20,33)(21,36,24,38)(27,43,35,42)(28,44,37,41)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293187);