#7719453

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

7025 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7626649);