#7626806

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×#537	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6848 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=g,e^2=h,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, 7534002);