#7626210

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	C₂²×#121	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7326 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)(5,6)(7,11)(8,9)(10,12)(13,14)(15,16)(17,25)(18,19)(20,21)(22,26)(23,27)(24,28);(6,9)(10,11)(14,19)(15,21)(22,25)(23,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,21)(14,20,19,16)(17,23,26,28)(22,27,25,24);(5,7,8,12)(6,10,9,11)(13,16,18,20)(14,21,19,15)(17,24,26,27)(22,28,25,23);(13,14)(15,20)(16,21)(17,22)(18,19)(23,27)(24,28)(25,26);(13,17)(14,22)(15,23)(16,24)(18,26)(19,25)(20,27)(21,28)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^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, 7533406);