#7627409

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

7036 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)(7,8)(9,10)(11,12)(13,14)(15,23)(16,18)(17,21)(19,20)(22,26)(24,28)(25,27);(6,10)(7,11)(14,20)(16,26)(21,23)(24,27);(5,7,9,11)(6,8,10,12)(13,15,19,17)(14,21,20,23)(16,24,26,27)(18,25,22,28);(5,8)(6,11)(7,10)(9,12)(13,16)(14,22)(15,24)(17,27)(18,20)(19,26)(21,28)(23,25);(1,2,3,4)(13,17,19,15)(14,21,20,23)(16,27,26,24)(18,25,22,28);(13,18,19,22)(14,16,20,26)(15,25,17,28)(21,24,23,27)

Polycyclic presentation:

2^9|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, 7534605);