#8037637

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

7340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336917
maximal normal subgroup of	#72336917
maximal quotient of	#72336975

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7944833);