#6963359

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2)(3,6)(4,5)(7,8)(11,17)(14,18)(19,20)(22,23)(25,26,29,34,30,33,43,49)(27,31,37,46,36,45,53,56)(28,38,41,48,40,50,42,32)(35,51,52,47,39,55,54,44);(1,3,7,4)(2,5,8,6)(9,10)(11,18)(12,15)(13,16)(14,17)(19,23)(20,22)(21,24)(25,27)(26,31)(28,39)(29,37)(30,36)(32,47)(33,45)(34,46)(35,40)(38,55)(41,54)(42,52)(43,53)(44,48)(49,56)(50,51);(1,4,7,3)(2,5,8,6)(9,11,13,17)(10,14,16,18)(12,20,21,19)(15,23,24,22)(25,28)(26,32)(27,35)(29,42)(30,40)(31,44)(33,48)(34,50)(36,39)(37,54)(38,49)(41,43)(45,47)(46,55)(51,56)(52,53);(1,5,7,6)(2,3,8,4)(9,12,13,21)(10,15,16,24)(11,19,17,20)(14,22,18,23)(27,36)(28,40)(31,45)(32,48)(37,53)(38,50)(41,42)(46,56);(1,6,7,5)(2,3,8,4)(10,16)(14,18)(15,24)(22,23)(26,33)(27,36)(32,48)(34,49)(35,39)(37,53)(38,50)(52,54)

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=fhi,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6870555);