#6817019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2150 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=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, 6724215);