#8461687

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6387 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=gh,e^2=ghi,f^2=gi,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, 8368883);