#366400

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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

1122 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273596);