#6996376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2413 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)(7,9)(8,10)(11,12)(13,14,17,22,18,23,31,19)(15,24,27,32,26,20,34,36)(16,21,28,38,30,39,42,33)(25,37,40,43,29,35,41,44);(5,6)(7,11)(8,10)(9,12)(14,19)(16,28)(17,31)(20,32)(22,23)(24,36)(25,40)(27,34)(29,41)(30,42)(33,38)(43,44);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,26)(14,20,23,24)(16,29,30,25)(17,27,31,34)(19,32,22,36)(21,37,39,35)(28,41,42,40)(33,44,38,43);(7,12)(9,11)(13,16,17,28,18,30,31,42)(14,21,22,38,23,39,19,33)(15,25,27,40,26,29,34,41)(20,35,36,44,24,37,32,43);(6,10)(9,11)(15,26)(20,24)(25,29)(27,34)(32,36)(35,37)(40,41)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6903572);