#7627445

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×#3566	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6500 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,e^2=i,f^2=gh,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, 7534641);