#8069527

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6061 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7976723);