#7634897

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
prime radical	C₂×#3455	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6782 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)(7,11)(8,16)(9,14)(10,13)(12,17)(15,20)(18,19)(21,22)(23,26)(24,31)(25,28)(27,33)(29,36)(30,35)(32,34);(2,4)(5,7,10,9)(6,11,13,14)(8,15,17,18)(12,19,16,20)(21,23,25,30)(22,26,28,35)(24,32,33,29)(27,36,31,34);(5,8,10,17)(6,12,13,16)(7,15,9,18)(11,19,14,20)(21,24,25,33)(22,27,28,31)(23,29,30,32)(26,34,35,36);(6,13)(8,17)(11,14)(15,18)(22,28)(26,35)(27,31)(34,36);(6,13)(11,14)(12,16)(19,20);(5,9,10,7)(6,14,13,11)(8,18,17,15)(12,20,16,19)

Polycyclic presentation:

2^9|b^2=h,c^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7542093);