#8755017

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3771 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)(3,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,2,7,4)(3,8,5,11)(6,12,14,9)(10,16,13,15)(18,23)(20,25)(24,29)(28,31)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(1,3,7,5)(2,8,4,11)(6,10,14,13)(9,15,12,16)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,4,7,2)(3,8,5,11)(6,12,14,9)(10,15,13,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32);(1,5,7,3)(2,8,4,11)(6,13,14,10)(9,15,12,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47);(1,6,7,14)(2,9,4,12)(3,10,5,13)(8,15,11,16)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=gi,e^2=gh,f^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662213);