#6467525

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2258 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374721);