#7641760

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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
prime radical	C₂×#3475	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6709 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7548956);