#229152

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	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

674 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,d^b=dei,e^b=eh,g^b=gh

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, 136348);