#8425608

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	#34681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(2,5)(6,7)(10,13)(11,15)(17,18,21,19)(20,23,24,22);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(18,19)(22,23);(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22);(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,22)(19,23)(21,24);(9,11,12,15)(10,14,13,16);(1,3)(2,6)(4,8)(5,7)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332804);