#7565493

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1813 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)(7,11)(8,12)(9,10)(13,14,17,19)(15,23,27,36)(16,28,31,39)(18,32,25,22)(20,35,33,24)(21,38,34,29)(26,40,30,37)(41,44,43,42);(1,2)(3,4)(6,10)(11,12)(14,19)(15,24)(16,29)(18,25)(20,36)(21,39)(23,33)(26,30)(27,35)(28,34)(31,38)(41,43);(5,7,9,8)(6,11,10,12)(13,15,25,35,17,27,18,24)(14,20,32,36,19,33,22,23)(16,30,38,42,31,26,29,44)(21,40,39,41,34,37,28,43);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,29,17,31,25,38)(14,21,22,28,19,34,32,39)(15,26,24,42,27,30,35,44)(20,37,23,41,33,40,36,43);(5,8,9,7)(6,11,10,12)(14,19)(15,27)(21,34)(22,32)(24,35)(26,30)(28,39)(42,44)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fgi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,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, 7472689);