#7656818

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

9018 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)(5,6,10,9)(7,12,8,11)(13,14,16,17)(15,19,20,18);(1,2)(3,4)(14,17)(18,19);(5,7,10,8)(6,11,9,12)(13,14)(15,18)(16,17)(19,20);(13,15)(14,18)(16,20)(17,19);(5,8,10,7)(6,11,9,12);(5,9,10,6)(7,12,8,11)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72266737, #72266997, #72266999

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564014);