#8092025

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

6411 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=h,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,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, 7999221);