#6467532

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2242 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=hi,d^2=h,e^2=gi,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, 6374728);