#7633926

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
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×#3444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6932 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)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22)(23,27)(24,28)(25,29)(26,30)(31,34)(32,35)(33,36);(2,4)(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,26,33)(22,27,30,36)(24,32,25,31)(28,35,29,34);(6,11)(10,16)(12,18)(17,20)(21,24,26,25)(22,28,30,29)(23,31,33,32)(27,34,36,35);(6,11)(7,13)(12,18)(14,19)(21,25,26,24)(22,29,30,28)(23,32,33,31)(27,35,36,34);(6,11)(10,16)(12,18)(17,20)(22,30)(27,36)(28,29)(34,35);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,e^a=ehi,c^b=ci,d^b=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, 7541122);