#7626569

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

8012 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^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, 7533765);