#7987197

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

6702 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)(7,13)(8,10)(9,12)(11,14)(15,20)(16,18)(17,19)(21,22,26,25)(23,29,31,27)(24,28,33,32)(30,35,36,34);(1,2)(3,4)(5,7)(6,11)(8,13)(9,17)(10,14)(12,20)(15,18)(16,19)(22,25)(27,29)(28,32)(34,35);(6,8)(7,14)(12,16)(15,17)(21,22,26,25)(23,27,31,29)(24,32,33,28)(30,35,36,34);(5,7)(6,11)(8,13)(9,15)(10,14)(12,19)(16,20)(17,18)(21,23,26,31)(22,27,25,29)(24,30,33,36)(28,34,32,35);(5,8,10,6)(7,13,14,11)(9,12,18,16)(15,19,17,20)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36);(5,9,10,18)(6,12,8,16)(7,15,14,17)(11,19,13,20)(21,25,26,22)(23,29,31,27)(24,28,33,32)(30,34,36,35)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=hi,e^2=h,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, 7894393);