#7819978

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

7808 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)(13,14,17,18)(15,19,20,16);(1,2,3,4)(5,6,8,9)(7,10,11,12)(15,20)(16,19);(6,9)(10,12)(13,15,17,20)(14,16,18,19);(7,11)(10,12)(13,14,17,18)(15,19,20,16);(5,7)(6,10)(8,11)(9,12)(13,14)(15,16)(17,18)(19,20);(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,19)(18,20)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7727174);