#7501770

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2387 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^c=ei,g^c=gi,g^d=gi

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, 7408966);