#7901881

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

6191 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257489, #72257577, #72264282
maximal normal subgroup of	#72257489, #72257577, #72264282
maximal quotient of	#72307809

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7809077);