#7969232

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

7901 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238572, #72238580, #72238582
maximal normal subgroup of	#72238572, #72238580, #72238582

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7876428);