#7627461

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×#3522	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6640 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=gh,f^2=h,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, 7534657);