#7985108

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

7711 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)(7,8)(9,11)(10,12)(13,14)(15,19)(16,18)(17,20);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,17)(16,20)(18,19);(5,8,10,9)(6,7,12,11)(13,14,16,18)(15,17,20,19);(5,7)(6,9)(8,12)(10,11)(13,14)(15,17)(16,18)(19,20);(5,7)(6,8)(9,12)(10,11)(13,15)(14,17)(16,20)(18,19);(5,9,10,8)(6,11,12,7)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,f^2=i,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	#72312421, #72315173, #72315175
maximal normal subgroup of	#72312421, #72315173, #72315175

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892304);