#7677050

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

7950 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)(8,12)(10,11)(13,14,17,16)(15,19,20,18);(5,6)(7,9)(8,10)(11,12)(13,15,17,20)(14,18,16,19);(2,4)(6,7)(10,11);(1,2)(3,4)(13,14,17,16)(15,18,20,19);(5,7,9,6)(8,11,12,10);(5,8)(6,10)(7,11)(9,12)(13,16,17,14)(15,19,20,18)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262487, #72262508, #72262517
maximal normal subgroup of	#72262487, #72262508, #72262517

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7584246);