#7738207

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5154 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7645403);