#7626589

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
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	C₂³×D₄	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6576 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)(7,14)(11,18)(15,19)(16,20)(22,24)(27,29)(28,32)(34,35);(5,6,10,13)(7,11,14,18)(8,9,12,17)(15,16,19,20)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(5,7)(6,11)(8,15)(9,16)(10,14)(12,19)(13,18)(17,20)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,8)(6,9)(7,15)(10,12)(11,16)(13,17)(14,19)(18,20)(21,24,26,22)(23,29,31,27)(25,32,33,28)(30,35,36,34);(5,9)(6,12)(7,16)(8,13)(10,17)(11,19)(14,20)(15,18)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35);(5,7)(6,11)(8,15)(9,16)(10,14)(12,19)(13,18)(17,20)

Polycyclic presentation:

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

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533785);