#7627263

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
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	C₂²×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 7534459);