#7627607

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3560	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6734 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534803);