#7689302

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	#36212	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6129 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596498);