#7917893

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

6887 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)(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18)(21,22)(23,30)(24,27)(25,32)(26,29)(28,34)(31,35)(33,36);(1,2)(3,4)(21,23)(22,26)(24,30)(25,31)(27,29)(28,35)(32,33)(34,36);(6,7)(11,13)(12,14)(18,19)(22,24)(23,29)(28,32)(31,36);(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18)(21,24,27,22)(23,30,29,26)(25,32,34,28)(31,33,36,35);(5,8)(6,11)(7,13)(9,15)(10,16)(12,18)(14,19)(17,20)(21,25)(22,28)(23,31)(24,32)(26,35)(27,34)(29,36)(30,33);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,26,27,30)(22,29,24,23)(25,33,34,35)(28,31,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7825089);