#8754830

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

5274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72266837, #72266838
Fitting subgroup of	#72266837
maximal subgroup of	#72213178, #72266837
maximal normal subgroup of	#72213178, #72266837
maximal quotient of	#72213086, #72213467, #72249412, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662026);