#7626330

order	512
nilpotency class	2
isoclinism	isoclinic class of #62817
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×#3562	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(1,2,4,3)(5,6)(7,9)(8,12)(10,11)(13,14,19,16)(15,18,23,20)(17,21,25,24)(22,26,28,27);(6,11)(7,12)(17,25)(21,24)(22,28)(26,27);(5,7,10,12)(6,8,11,9)(13,15,19,23)(14,20,16,18)(17,22,25,28)(21,27,24,26);(1,2,4,3)(5,8,10,9)(6,7,11,12)(13,16,19,14)(15,18,23,20)(17,24,25,21)(22,26,28,27);(1,3,4,2)(5,6)(7,8)(9,12)(10,11)(13,17,19,25)(14,21,16,24)(15,22,23,28)(18,26,20,27);(5,9,10,8)(6,12,11,7)(13,18)(14,15)(16,23)(17,26)(19,20)(21,22)(24,28)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533526);