#7626552

order	512
nilpotency class	2
isoclinism	isoclinic class of #62862
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×#3561	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6850 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)(7,11)(8,10)(9,12)(13,14,18,15)(16,19,23,21)(17,22,25,20)(24,27,28,26);(5,6,8,10)(7,9,12,11)(16,23)(19,21)(24,28)(26,27);(5,6)(7,9)(8,10)(11,12)(14,15)(19,21)(20,22)(26,27);(5,7,8,12)(6,9,10,11)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(1,3,4,2)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27);(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533748);