#7864177

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5558 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)(3,8)(4,6)(5,7)(9,10,13,15)(11,12,14,16)(18,21)(22,24);(1,3,5,6)(2,4,7,8)(9,11,13,14)(10,12,15,16)(17,18)(19,22)(20,21)(23,24);(1,2)(3,8)(4,6)(5,7)(9,10)(11,16)(12,14)(13,15)(17,19,20,23)(18,22,21,24);(9,11,13,14)(10,12,15,16);(1,2)(3,4)(5,7)(6,8)(9,10)(11,12)(13,15)(14,16);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,15)(13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304331, #72304387, #72304490
maximal normal subgroup of	#72304331, #72304387, #72304490

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7771373);