#7055878

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1985 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)(7,9)(8,10)(11,12)(13,14,18,24,19,23,34,40)(15,25,28,38,27,42,31,20)(16,29,33,44,32,21,39,37)(17,22,26,41,35,36,30,43);(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,20,23,38)(16,30,32,26)(17,33,35,39)(18,31,34,28)(21,22,29,36)(24,42,40,25)(37,43,44,41);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,32)(14,21,23,29)(15,26,27,30)(17,28,35,31)(18,33,34,39)(20,36,38,22)(24,37,40,44)(25,43,42,41);(7,12)(9,11)(13,17,34,30,19,35,18,26)(14,22,40,43,23,36,24,41)(15,16,31,39,27,32,28,33)(20,37,42,21,38,44,25,29);(6,10)(9,11)(14,23)(15,27)(16,32)(22,36)(24,40)(28,31)(33,39)(41,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=fi,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=dh,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, 6963074);