#7634919

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,e^a=ehi,c^b=ci,d^c=dhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303354
maximal normal subgroup of	#72303354

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542115);