#7672748

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×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579944);