#7987161

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

6821 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)(7,15)(8,11)(9,18)(10,14)(12,16)(13,20)(17,19)(21,22,26,25)(23,29,31,27)(24,28,33,32)(30,35,36,34);(1,2)(3,4)(5,7)(6,12)(8,15)(9,17)(10,20)(11,16)(13,19)(14,18)(22,25)(27,29)(28,32)(34,35);(6,8)(7,16)(9,20)(14,19)(21,22,26,25)(23,27,31,29)(24,32,33,28)(30,35,36,34);(5,8,11,6)(7,15,16,12)(9,18,20,13)(10,19,17,14)(21,23,26,31)(22,27,25,29)(24,30,33,36)(28,34,32,35);(5,9,11,20)(6,13,8,18)(7,17,16,10)(12,19,15,14)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36);(5,10,11,17)(6,14,8,19)(7,9,16,20)(12,13,15,18)(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=i,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, 7894357);