#8001201

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

7575 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)(13,14,16,19)(15,17,20,18);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,16,20)(14,17,19,18);(5,6,8,10)(7,9,11,12)(13,15)(14,18)(16,20)(17,19);(7,11)(9,12)(13,15,16,20)(14,17,19,18);(5,7)(6,9)(8,11)(10,12)(13,15)(14,17)(16,20)(18,19);(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,19)(18,20)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7908397);