#7904008

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

6771 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)(5,6,9,10)(7,13,15,11)(8,12,17,18)(14,20,16,19)(24,32)(27,34)(31,36)(33,35);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,20)(17,18)(21,22)(23,28)(24,27)(25,29)(26,30)(31,35)(32,34)(33,36);(6,10)(11,13)(12,18)(19,20)(21,23)(22,25)(24,31)(26,29)(27,33)(28,30)(32,36)(34,35);(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,20)(17,18)(21,24)(22,27)(23,31)(25,33)(26,32)(28,35)(29,36)(30,34);(5,7,9,15)(6,11,10,13)(8,16,17,14)(12,20,18,19)(21,23)(22,28)(24,31)(25,30)(26,29)(27,35)(32,36)(33,34);(5,8,9,17)(6,12,10,18)(7,14,15,16)(11,19,13,20)(21,25,26,28)(22,29,30,23)(24,33,32,35)(27,36,34,31)

Polycyclic presentation:

2^9|a^2=h,e^2=h,f^2=i,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, 7811204);