#8006496

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

7249 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)(5,6,10,11)(7,12,14,8)(9,17,18,13)(15,16,20,19)(21,22)(23,27)(24,32)(25,26)(28,33)(29,31)(30,35)(34,36);(1,2)(3,4)(6,11)(8,12)(13,17)(16,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(6,11)(7,14)(13,17)(15,20)(22,26)(24,33)(27,29)(30,36);(5,7)(6,12)(8,11)(9,15)(10,14)(13,19)(16,17)(18,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(5,8,10,12)(6,7,11,14)(9,16,18,19)(13,15,17,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36);(5,9)(6,13)(7,15)(8,16)(10,18)(11,17)(12,19)(14,20)(21,24)(22,28)(23,30)(25,33)(26,32)(27,34)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=hi,e^2=hi,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336756
maximal normal subgroup of	#72336756
maximal quotient of	#72336920

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7913692);