#6816833

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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×#3273	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2174 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)(5,6,9,11)(7,12,8,10)(13,14,17,21)(15,23,26,19)(16,20,30,36)(18,22,25,32)(24,37,38,31)(27,35,29,33)(28,34,40,43)(39,44,41,42);(5,6)(7,10)(8,12)(9,11)(15,24)(16,28)(18,25)(19,31)(20,34)(22,32)(23,37)(26,38)(27,29)(30,40)(33,35)(36,43);(1,2,3,4)(13,15,25,38,17,26,18,24)(14,19,32,37,21,23,22,31)(16,29,40,39,30,27,28,41)(20,35,43,42,36,33,34,44);(5,7,9,8)(6,10,11,12)(15,26)(19,23)(24,38)(27,29)(31,37)(33,35)(39,41)(42,44);(5,8,9,7)(6,10,11,12)(13,16,25,40,17,30,18,28)(14,20,32,43,21,36,22,34)(15,27,38,41,26,29,24,39)(19,33,37,44,23,35,31,42)

Polycyclic presentation:

2^9|a^2=fi,c^2=gh,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6724029);