#6683003

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2018 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6590199);