#8455732

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3869 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=h,e^2=hi,f^2=gh,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, 8362928);