#7739556

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

7480 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336917
maximal normal subgroup of	#72336917
maximal quotient of	#72336951

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7646752);