#7671279

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

7868 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305378
maximal normal subgroup of	#72305378
maximal quotient of	#72306262, #72306264

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7578475);