#6993026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37422	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1612 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,28,37)(20,25,32,42)(21,24,34,27)(29,36,31,40)(30,38,43,35)(33,39,46,41)(44,48,45,47);(1,3,7,4)(2,5,8,6)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,31)(21,30)(22,28)(24,38)(25,40)(26,37)(27,35)(29,32)(33,45)(34,43)(36,42)(39,47)(41,48)(44,46);(1,4,7,3)(2,5,8,6)(11,16)(13,15)(18,24)(19,28)(20,32)(21,34)(23,35)(25,41)(26,27)(36,48)(37,38)(39,42)(40,47)(44,45);(1,5,7,6)(2,3,8,4)(10,14)(13,15)(17,20)(18,25)(19,29)(21,33)(22,32)(23,36)(24,39)(26,42)(27,41)(28,31)(30,44)(34,46)(35,47)(37,40)(38,48)(43,45);(1,6,7,5)(2,3,8,4)(18,26)(19,28)(20,32)(24,27)(30,43)(33,46)(36,40)(47,48)

Polycyclic presentation:

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

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, 6900222);