#7627591

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×#3548	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6872 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)(5,6,11,10)(7,14,9,12)(8,16,18,13)(15,19,17,20)(21,22)(23,28)(24,31)(25,30)(26,32)(27,33)(29,34)(35,36);(5,7,11,9)(6,12,10,14)(8,17,18,15)(13,20,16,19)(21,23)(22,28)(24,30)(25,31)(26,34)(27,35)(29,32)(33,36);(5,7,11,9)(6,12,10,14)(8,15,18,17)(13,19,16,20)(21,24)(22,29)(23,26)(25,28)(27,34)(30,35)(31,33)(32,36);(5,8,11,18)(6,13,10,16)(7,15,9,17)(12,19,14,20)(21,25,27,32)(22,30,33,26)(23,29,35,31)(24,28,34,36);(1,3,4,2)(5,9,11,7)(6,12,10,14)(8,17,18,15)(13,19,16,20)(21,24)(22,31)(23,30)(25,28)(26,35)(27,34)(29,33)(32,36);(5,10,11,6)(7,14,9,12)(8,16,18,13)(15,20,17,19)(21,26,27,30)(22,32,33,25)(23,34,35,24)(28,29,36,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,e^2=i,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, 7534787);