#7917897

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

7018 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)(23,25)(24,27)(26,29)(28,30)(31,33)(32,35)(34,36);(1,2)(3,4)(21,23)(22,25)(24,30)(26,31)(27,28)(29,33)(32,34)(35,36);(6,7)(11,13)(12,14)(18,19)(22,24)(23,28)(29,32)(31,36);(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18)(21,24,27,22)(23,30,28,25)(26,32,35,29)(31,34,36,33);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,25,27,30)(22,28,24,23)(26,34,35,33)(29,31,32,36);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,26,27,35)(22,29,24,32)(23,31,28,36)(25,33,30,34)

Polycyclic presentation:

2^9|a^2=hi,d^2=h,e^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, 7825093);