#6373425

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2123 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,e^b=ei,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, 6280621);