#7912288

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

6905 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)(6,10)(7,13)(12,18)(14,15)(21,22)(23,26)(24,30)(25,33)(27,29)(28,32)(31,34)(35,36);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,17)(14,19)(15,20)(16,18)(21,23,28,27)(22,29,32,26)(24,33,35,31)(25,30,34,36);(5,6,9,10)(7,11,13,17)(8,12,16,18)(14,19,15,20)(21,24,28,35)(22,30,32,36)(23,31,27,33)(25,26,34,29);(5,6)(7,11)(8,12)(9,10)(13,17)(14,19)(15,20)(16,18)(21,25,28,34)(22,31,32,33)(23,30,27,36)(24,26,35,29);(5,7,9,13)(6,11,10,17)(8,15,16,14)(12,20,18,19)(21,26)(22,23)(24,34)(25,35)(27,32)(28,29)(30,31)(33,36);(5,8,9,16)(6,12,10,18)(7,14,13,15)(11,19,17,20)(21,27,28,23)(22,29,32,26)(24,33,35,31)(25,36,34,30)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=hi,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, 7819484);