#7625900

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×H(5,𝔽₂)
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×#3480	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7426 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)(6,10)(11,12)(13,14,20,23)(15,24,26,21)(16,27,17,19)(18,22,28,25);(5,6,9,10)(7,11,8,12)(13,15)(14,21)(16,25)(17,22)(18,19)(20,26)(23,24)(27,28);(13,16,20,17)(14,19,23,27)(15,25,26,22)(18,24,28,21);(5,7,9,8)(6,11,10,12)(13,17,20,16)(14,19,23,27)(15,22,26,25)(18,24,28,21);(13,18)(14,22)(15,19)(16,24)(17,21)(20,28)(23,25)(26,27);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,19)(14,16)(15,18)(17,23)(20,27)(21,25)(22,24)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533096);