#7680638

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

7417 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=h,e^2=h,f^2=hi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7587834);