#6396303

order	512
nilpotency class	3
isoclinism	isoclinic class of #62265
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 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)(15,24)(16,21)(17,22)(18,33)(19,23)(20,27)(25,40)(26,35)(28,41)(29,44)(30,37)(31,39)(32,38)(34,42)(36,43);(1,2,4,3)(6,9)(7,11);(1,3,4,2)(13,15,18,27,19,28,17,26)(14,20,22,24,23,35,33,41)(16,29,30,43,31,25,32,42)(21,36,38,44,39,34,37,40);(5,7,8,11)(6,10,9,12)(13,16,19,31)(14,21,23,39)(15,25,28,29)(17,32,18,30)(20,34,35,36)(22,38,33,37)(24,40,41,44)(26,43,27,42);(6,9)(10,12)(13,17,19,18)(14,22,23,33)(15,26,28,27)(16,30,31,32)(20,24,35,41)(21,37,39,38)(25,42,29,43)(34,44,36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,d^c=di,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6303499);