#7720528

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

6560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,e^2=h,f^2=i,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, 7627724);