#7920125

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

6882 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)(7,11)(8,16)(9,14)(10,12)(13,18)(15,20)(17,19)(21,22,26,28)(23,25,30,27)(24,32,34,29)(31,33,36,35);(1,2)(3,4)(5,7,10,9)(6,11,12,14)(8,15,18,17)(13,19,16,20)(21,23,26,30)(22,27,28,25)(24,31,34,36)(29,33,32,35);(6,12)(11,14)(13,16)(19,20)(22,28)(23,30)(29,32)(31,36);(5,6)(7,11)(8,13)(9,14)(10,12)(15,19)(16,18)(17,20)(21,22)(23,27)(24,29)(25,30)(26,28)(31,33)(32,34)(35,36);(5,8)(6,13)(7,15)(9,17)(10,18)(11,19)(12,16)(14,20)(21,24)(22,29)(23,31)(25,35)(26,34)(27,33)(28,32)(30,36);(5,9,10,7)(6,14,12,11)(8,17,18,15)(13,20,16,19)(21,25,26,27)(22,30,28,23)(24,33,34,35)(29,31,32,36)

Polycyclic presentation:

2^9|a^2=i,b^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, 7827321);