#221503

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×C₂²⋊C₄
prime radical	#2407	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1137 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)(5,6,9,12,10,8,7,11)(13,14)(15,19)(16,23)(17,26)(18,21)(20,22)(24,28)(25,27);(1,2)(3,4)(5,7,10,9)(6,11,8,12)(13,15,18,17)(14,19,21,26)(16,24,25,22)(20,23,28,27);(5,8,10,6)(7,12,9,11)(13,16,18,25)(14,20,21,28)(15,22,17,24)(19,27,26,23);(2,4)(5,7,10,9)(6,11,8,12)(13,15,18,17)(14,19,21,26)(16,22,25,24)(20,27,28,23)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=chi,d^b=de,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128699);