#8456957

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×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4088 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8364153);