#7627397

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
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×#532	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6948 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)(5,6,10,9)(7,12,8,11)(14,15)(19,21)(20,22)(27,28);(5,6,10,9)(7,12,8,11)(13,14,18,15)(16,19,24,21)(17,20,26,22)(23,27,25,28);(1,2,4,3)(5,7,10,8)(6,11,9,12)(13,14,18,15)(16,19,24,21)(17,20,26,22)(23,27,25,28);(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27);(1,3,4,2)(5,8,10,7)(6,11,9,12)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27);(5,9,10,6)(7,12,8,11)(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534593);