#7808374

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

7344 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)(6,11)(7,13)(12,16)(15,19)(22,24)(23,29)(28,32)(31,35);(1,2)(3,4)(5,6)(7,14)(8,16)(9,13)(10,11)(12,18)(15,17)(19,20)(22,24)(23,29)(25,33)(34,36);(6,11)(9,14)(12,16)(17,20)(21,22,26,24)(23,30,29,27)(25,32,33,28)(31,34,35,36);(5,7)(6,9)(8,15)(10,13)(11,14)(12,17)(16,20)(18,19)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36);(5,8)(6,12)(7,15)(9,17)(10,18)(11,16)(13,19)(14,20)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34);(5,9)(6,7)(8,17)(10,14)(11,13)(12,15)(16,19)(18,20)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

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

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	#72336964, #72336969

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7715570);