#6373098

order	512
nilpotency class	3
isoclinism	isoclinic class of #60372
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280294);