#7020706

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45236	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,25)(21,26)(22,23)(24,33)(27,29)(30,45)(31,38)(32,43)(34,39)(35,48)(36,42)(37,44)(40,46)(41,47);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(18,23)(20,34)(24,38)(26,27)(28,43)(30,35)(36,37)(46,47);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,29,31,22,32,21,33)(18,24,26,43,23,38,27,28)(20,35,36,46,34,30,37,47)(25,41,44,45,39,40,42,48);(3,8)(6,7)(11,16)(13,15)(17,20,22,34)(18,25,23,39)(19,30,32,35)(21,37,29,36)(24,40,38,41)(26,44,27,42)(28,45,43,48)(31,47,33,46);(10,14)(13,15)(17,21,22,29)(18,26,23,27)(19,31,32,33)(20,36,34,37)(24,28,38,43)(25,42,39,44)(30,46,35,47)(40,48,41,45)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fg,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=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, 6927902);