#8091690

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

6694 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)(5,6,11,13)(7,12,8,10)(9,14,18,20)(15,19,16,17)(21,22,27,28)(23,25,24,29)(26,30,34,35)(31,33,32,36);(1,2)(3,4)(5,7,11,8)(6,12,13,10)(9,15,18,16)(14,19,20,17)(22,28)(23,24)(30,35)(31,32);(6,13)(10,12)(14,20)(17,19)(21,23,27,24)(22,29,28,25)(26,32,34,31)(30,33,35,36);(5,8,11,7)(6,10,13,12)(9,15,18,16)(14,19,20,17)(21,24,27,23)(22,29,28,25)(26,31,34,32)(30,33,35,36);(5,9,11,18)(6,14,13,20)(7,15,8,16)(10,17,12,19)(21,25,27,29)(22,23,28,24)(26,33,34,36)(30,31,35,32);(5,10)(6,7)(8,13)(9,17)(11,12)(14,15)(16,20)(18,19)(21,26,27,34)(22,30,28,35)(23,31,24,32)(25,33,29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=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, 7998886);