#8430425

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	#41468	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4181 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,e^2=g,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, 8337621);