#7626774

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

6376 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)(5,6,11,15)(7,14,10,12)(8,9,13,20)(16,19,18,17)(21,22)(23,25)(24,32)(26,30)(27,28)(29,35)(31,34)(33,36);(1,3,4,2)(22,28)(23,30)(29,32)(31,36);(5,7,11,10)(6,12,15,14)(8,16,13,18)(9,17,20,19)(21,23,27,30)(22,25,28,26)(24,31,35,36)(29,33,32,34);(5,8)(6,9)(7,16)(10,18)(11,13)(12,17)(14,19)(15,20)(21,24,27,35)(22,29,28,32)(23,31,30,36)(25,33,26,34);(5,9)(6,13)(7,17)(8,15)(10,19)(11,20)(12,18)(14,16)(21,25,27,26)(22,23,28,30)(24,33,35,34)(29,31,32,36);(5,10,11,7)(6,14,15,12)(8,18,13,16)(9,19,20,17)(21,26,27,25)(22,30,28,23)(24,34,35,33)(29,36,32,31)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=g,e^2=g,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, 7533970);