#8093610

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263055, #72263064
maximal normal subgroup of	#72263055, #72263064

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000806);