#7626445

order	512
nilpotency class	2
isoclinism	isoclinic class of #62862
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	C₂²×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6446 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)(5,6,9,10)(7,11,15,13)(8,16,17,12)(14,19,20,18)(22,26)(24,32)(27,29)(30,36);(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(1,3,4,2)(6,10)(11,13)(12,16)(18,19)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(5,6)(7,11)(8,12)(9,10)(13,15)(14,18)(16,17)(19,20)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20);(5,8,9,17)(6,12,10,16)(7,14,15,20)(11,18,13,19)

Polycyclic presentation:

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

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, 7533641);