#7627642

order	512
nilpotency class	2
isoclinism	isoclinic class of #62972
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×#538	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6234 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534838);