#7634913

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

6606 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)(5,6,10,13)(7,14,16,11)(8,17,9,12)(15,19,18,20)(21,22,25,27)(23,26,29,34)(24,31,32,28)(30,36,35,33);(1,2,3,4)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35);(5,7)(6,11)(8,18)(9,15)(10,16)(12,20)(13,14)(17,19)(22,27)(23,29)(28,31)(30,35);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,25,32)(22,28,27,31)(23,30,29,35)(26,33,34,36);(6,13)(11,14)(12,17)(19,20);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^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, 7542109);