#7631160

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

8828 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)(5,6,11,9)(7,8,12,10)(13,14,16,18)(15,17,19,20);(6,9)(7,12)(13,14)(15,17)(16,18)(19,20);(1,2)(3,4)(5,7)(6,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20);(5,8,11,10)(6,12,9,7);(5,9,11,6)(7,8,12,10);(5,10,11,8)(6,12,9,7)(13,15,16,19)(14,17,18,20)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304616, #72304618, #72304631
maximal normal subgroup of	#72304616, #72304618, #72304631

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538356);