#7645840

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
prime radical	#37536	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6648 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)(6,8)(7,13)(12,16)(15,19)(22,26)(23,29)(28,33)(30,35);(1,2)(3,4)(5,6,10,8)(7,11,13,14)(9,12,17,16)(15,18,19,20)(22,26)(27,32)(28,33)(34,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(5,8,10,6)(7,14,13,11)(9,12,17,16)(15,18,19,20)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553036);