#7627419

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×#3531	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6294 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^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, 7534615);