#7627534

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×#3559	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=g,e^2=hi,f^2=gh,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, 7534730);