#7627674

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×#3568	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6416 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)(23,26)(24,31)(25,28)(27,32)(29,33)(30,36)(34,35);(7,14)(8,16)(11,13)(12,17)(21,22,25,28)(23,26,29,33)(24,27,32,31)(30,34,35,36);(6,10)(11,13)(12,17)(18,19)(23,29)(26,33)(30,35)(34,36);(5,7)(6,11)(8,15)(9,14)(10,13)(12,18)(16,20)(17,19)(21,23,25,29)(22,26,28,33)(24,30,32,35)(27,34,31,36);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36);(5,8,9,16)(6,12,10,17)(7,15,14,20)(11,18,13,19)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,f^2=gh,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, 7534870);