#7627406

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×#3529	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7108 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:

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

Polycyclic presentation:

2^9|a^2=g,b^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, 7534602);