#7904587

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

7290 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)(5,6,10,13)(7,8,15,11)(9,12,17,19)(14,16,20,18)(21,22,25,28)(23,29,31,26)(24,27,33,32)(30,35,36,34);(1,2)(3,4)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,7,10,15)(6,11,13,8)(9,14,17,20)(12,18,19,16)(21,23,25,31)(22,26,28,29)(24,30,33,36)(27,34,32,35);(5,7)(6,8)(9,14)(10,15)(11,13)(12,16)(17,20)(18,19)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,8)(6,7)(9,16)(10,11)(12,14)(13,15)(17,18)(19,20)(21,22)(23,26)(24,32)(25,28)(27,33)(29,31)(30,35)(34,36);(5,9,10,17)(6,12,13,19)(7,14,15,20)(8,16,11,18)(21,24)(22,27)(23,30)(25,33)(26,34)(28,32)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=h,c^2=h,f^2=hi,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.

Relationships

maximal subgroup of	#72336756, #72336917
maximal normal subgroup of	#72336756, #72336917
maximal quotient of	#72336930, #72336959

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7811783);