#7715003

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

7075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

(1,2,3,4)(5,6,10,11)(7,8,13,16)(9,12,17,18)(14,15,19,20)(25,33)(28,32)(30,36)(34,35);(1,2)(3,4)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(6,11)(7,13)(12,18)(14,19)(22,24)(27,29)(28,32)(34,35);(5,7)(6,8)(9,14)(10,13)(11,16)(12,15)(17,19)(18,20)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,8)(6,7)(9,15)(10,16)(11,13)(12,14)(17,20)(18,19)(21,24,26,22)(23,29,31,27)(25,32,33,28)(30,35,36,34);(5,9,10,17)(6,12,11,18)(7,14,13,19)(8,15,16,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622199);