#7627544

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×#3558	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7298 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^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, 7534740);