#7627210

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	C₂×#3541	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6728 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)(6,11)(7,13)(9,17)(16,19)(22,25)(27,30)(28,33)(35,36);(1,2,4,3)(5,6)(7,12)(8,13)(9,14)(10,11)(15,19)(16,20)(17,18);(5,7)(6,8)(9,15)(10,13)(11,12)(14,16)(17,20)(18,19)(21,22,26,25)(23,27,31,30)(24,28,34,33)(29,35,32,36);(5,7)(6,12)(8,11)(9,15)(10,13)(14,19)(16,18)(17,20)(21,23,26,31)(22,27,25,30)(24,32,34,29)(28,36,33,35);(1,3,4,2)(5,8,10,12)(6,13,11,7)(9,16,17,19)(14,20,18,15)(21,24,26,34)(22,28,25,33)(23,29,31,32)(27,35,30,36);(5,9)(6,14)(7,15)(8,16)(10,17)(11,18)(12,19)(13,20)(21,25,26,22)(23,30,31,27)(24,33,34,28)(29,36,32,35)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=gh,e^2=hi,f^2=gh,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, 7534406);