#7626848

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×#3485	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8336 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)(6,9)(10,12)(13,14,20,18)(15,24,23,19)(16,26,28,21)(17,25,27,22);(13,15)(14,19)(16,22)(17,21)(18,24)(20,23)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,25)(17,24)(18,26)(19,27)(20,28)(22,23);(7,11)(10,12)(13,17,20,27)(14,22,18,25)(15,21,23,26)(16,24,28,19);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,18,20,14)(15,24,23,19)(16,26,28,21)(17,22,27,25);(5,6)(7,10)(8,9)(11,12)(13,19,20,24)(14,23,18,15)(16,27,28,17)(21,22,26,25)

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=i,f^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, 7534044);