#225403

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂²×D₄	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ch,d^b=dei,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132599);