#7627443

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
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×#3547	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6846 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)(7,8)(9,10)(11,12)(13,14)(15,16)(17,21)(18,22)(19,23)(20,26)(24,25)(27,28);(6,10)(7,11)(13,15)(14,16)(17,18)(19,20)(21,22)(23,26)(24,27)(25,28);(5,7,9,11)(6,8,10,12)(13,16)(14,20)(15,23)(17,25)(18,21)(19,26)(22,27)(24,28);(5,8)(6,11)(7,10)(9,12)(13,17)(14,21)(15,24)(16,25)(18,20)(19,27)(22,26)(23,28);(1,3,4,2)(13,16)(14,15)(17,25)(18,28)(19,26)(20,23)(21,24)(22,27);(13,18,19,24)(14,22,23,25)(15,17,20,27)(16,21,26,28)

Polycyclic presentation:

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

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, 7534639);