#7627471

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×#3522	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7000 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)(7,11)(8,10)(9,12)(13,14)(15,23)(16,26)(17,20)(18,22)(19,21)(24,25)(27,28);(5,6,8,10)(7,9,12,11)(13,15,20,19)(14,21,17,23)(16,24,22,27)(18,25,26,28);(5,6)(7,9)(8,10)(11,12)(13,16)(14,18)(15,24)(17,26)(19,27)(20,22)(21,25)(23,28);(5,7,8,12)(6,9,10,11)(13,17,20,14)(15,21,19,23)(16,26,22,18)(24,25,27,28);(13,18,20,26)(14,22,17,16)(15,25,19,28)(21,27,23,24);(13,19,20,15)(14,21,17,23)(16,27,22,24)(18,25,26,28)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=h,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, 7534667);