#6506708

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500443
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3063 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6413904);