#7627405

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×#3529	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6818 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=i,f^2=g,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, 7534601);