#7664939

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

7889 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)(7,8)(9,11)(10,12)(14,17)(15,19);(1,2)(3,4)(14,17)(18,20);(5,7)(6,10)(8,11)(9,12)(13,14,16,17)(15,18,19,20);(5,8,9,10)(6,7,11,12)(13,15,16,19)(14,18,17,20);(5,6)(7,8)(9,11)(10,12);(5,7)(6,10)(8,11)(9,12)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262515, #72262546, #72262558
maximal normal subgroup of	#72262515, #72262546, #72262558

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572135);