#7626678

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

6792 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)(5,6,11,15)(7,10,9,12)(8,14,19,13)(16,20,17,18)(21,22)(23,30)(24,27)(25,33)(26,29)(28,34)(31,32)(35,36);(1,3,4,2)(5,7,11,9)(6,12,15,10)(8,17,19,16)(13,18,14,20);(5,8)(6,13)(7,16)(9,17)(10,18)(11,19)(12,20)(14,15)(21,23,26,27)(22,24,29,30)(25,31,34,35)(28,32,33,36);(5,9,11,7)(6,10,15,12)(8,17,19,16)(13,18,14,20)(21,22)(23,24)(25,28)(26,29)(27,30)(31,32)(33,34)(35,36);(5,8)(6,14)(7,17)(9,16)(10,18)(11,19)(12,20)(13,15)(21,24)(22,27)(23,29)(25,32)(26,30)(28,35)(31,33)(34,36);(5,10)(6,7)(8,18)(9,15)(11,12)(13,16)(14,17)(19,20)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^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, 7533874);