#7065139

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1811 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)(13,14,19,20,18,23,17,24)(15,25,32,37,27,21,28,34)(16,29,33,38,31,22,30,35)(26,36,42,43,41,39,40,44);(5,6)(7,11)(8,10)(9,12)(14,20)(16,30)(17,19)(21,34)(23,24)(25,37)(26,40)(28,32)(31,33)(35,38)(41,42)(43,44);(2,4)(5,7,8,12)(6,9,10,11)(13,15,18,27)(14,21,23,25)(16,26,31,41)(17,28,19,32)(20,34,24,37)(22,36,29,39)(30,40,33,42)(35,43,38,44);(7,12)(9,11)(13,16,17,30,18,31,19,33)(14,22,24,38,23,29,20,35)(15,26,28,40,27,41,32,42)(21,36,37,44,25,39,34,43);(6,10)(9,11)(14,23)(15,27)(20,24)(22,29)(26,41)(28,32)(35,38)(40,42)

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6972335);