#7626262

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×#3514	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7310 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)(14,18)(19,24)(20,25)(26,28);(6,11)(7,12)(16,23)(20,25)(21,27)(26,28);(1,3,4,2)(5,7,10,12)(6,8,11,9)(13,14)(15,19)(16,20)(17,18)(21,26)(22,24)(23,25)(27,28);(5,8,10,9)(6,7,11,12)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28);(5,6)(7,8)(9,12)(10,11)(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,27)(24,28);(5,9,10,8)(6,12,11,7)(13,14)(15,19)(16,20)(17,18)(21,26)(22,24)(23,25)(27,28)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=h,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, 7533458);