#7627140

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×#121	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6266 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)(5,6,11,9)(7,14,10,12)(8,13,17,18)(15,20,16,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36);(1,2,4,3)(22,26)(27,29)(28,33)(34,35);(5,7,11,10)(6,12,9,14)(8,15,17,16)(13,19,18,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(5,8)(6,13)(7,15)(9,18)(10,16)(11,17)(12,19)(14,20)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35);(1,3,4,2)(5,9,11,6)(7,14,10,12)(8,13,17,18)(15,19,16,20);(5,10,11,7)(6,14,9,12)(8,16,17,15)(13,20,18,19)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=g,e^2=i,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, 7534336);