#7938633

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

6734 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)(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18)(21,22,27,28)(23,29,24,26)(25,33,35,30)(31,36,32,34);(1,2)(3,4)(21,23,27,24)(22,26,28,29)(25,31,35,32)(30,36,33,34);(6,7)(11,13)(12,14)(18,19)(22,28)(23,24)(30,33)(31,32);(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18)(21,24,27,23)(22,29,28,26)(25,32,35,31)(30,34,33,36);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,25,27,35)(22,30,28,33)(23,31,24,32)(26,36,29,34);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,26,27,29)(22,23,28,24)(25,34,35,36)(30,32,33,31)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,e^2=h,f^2=h,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, 7845829);