#7672687

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

8112 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=h,e^2=i,f^2=i,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	#72315184, #72315202
maximal normal subgroup of	#72315184, #72315202

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579883);