#6395827

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1881 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)(8,11)(9,12)(13,14)(15,19)(16,21)(17,20)(18,23)(22,30)(24,32)(25,31)(26,34)(27,36)(28,37)(29,35)(33,40)(38,42)(39,43)(41,44);(1,2,4,3)(5,7,10,6)(8,11,9,12)(13,15,18,26)(14,19,23,34)(16,27,28,24)(17,25,30,40)(20,31,22,33)(21,36,37,32)(29,41,42,39)(35,44,38,43);(1,3,4,2)(14,20)(17,30)(19,31)(21,35)(22,23)(25,40)(29,42)(32,43)(33,34)(36,44)(37,38)(39,41);(5,8,10,9)(6,11,7,12)(13,16)(14,21)(15,24)(17,29)(18,28)(19,32)(20,35)(22,38)(23,37)(25,39)(26,27)(30,42)(31,43)(33,44)(34,36)(40,41);(5,9,10,8)(6,11,7,12)(16,28)(21,37)(24,27)(29,42)(32,36)(35,38)(39,41)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6303023);