#7632506

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

7871 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 20 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284872, #72285009
maximal normal subgroup of	#72284872, #72285009

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539702);