#7916683

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

7232 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)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(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);(6,10)(11,13)(12,17)(18,19)(21,23,25,31)(22,26,28,29)(24,30,33,36)(27,34,32,35);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,22)(23,26)(24,32)(25,28)(27,33)(29,31)(30,35)(34,36);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,24)(22,27)(23,30)(25,33)(26,34)(28,32)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=i,c^2=i,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
maximal normal subgroup of	#72336756
maximal quotient of	#72336838

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7823879);