#7699749

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×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5706 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,e^2=gh,b^a=bg,c^a=ch,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304364, #72304396, #72304492
maximal normal subgroup of	#72304364, #72304396, #72304492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606945);