#6910245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#37683	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1824 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6817441);