#8695328

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
prime radical	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31);(2,7)(3,10)(9,16)(11,13)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(34,39)(35,41)(40,47)(42,48);(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,34,38,39)(35,37,41,46)(36,40,44,47)(42,45,48,43);(1,3)(2,8)(4,7)(5,13)(6,10)(9,12)(11,15)(14,16)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,35)(34,37)(36,42)(38,41)(39,46)(40,45)(43,47)(44,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,13,16,11)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,5,6,15)(2,9,7,16)(3,11,10,13)(4,12,8,14)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37)(34,41)(35,39)(36,43)(38,46)(40,42)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8602524);