#7988220

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

6913 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)(7,14)(12,18)(15,20)(16,19)(22,26)(23,29)(28,32)(31,35);(1,2)(3,4)(5,6)(7,12)(8,13)(9,17)(10,11)(14,18)(15,19)(16,20)(21,22)(23,27)(24,32)(25,26)(28,33)(29,30)(31,36)(34,35);(6,11)(8,17)(12,18)(15,20)(21,22,25,26)(23,30,29,27)(24,28,33,32)(31,36,35,34);(5,7)(6,12)(8,15)(9,16)(10,14)(11,18)(13,19)(17,20)(21,23,25,29)(22,27,26,30)(24,31,33,35)(28,34,32,36);(5,8,10,17)(6,13,11,9)(7,15,14,20)(12,19,18,16)(21,22)(23,27)(24,32)(25,26)(28,33)(29,30)(31,36)(34,35);(5,9,10,13)(6,8,11,17)(7,16,14,19)(12,15,18,20)(21,24,25,33)(22,28,26,32)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7895416);