#202408

order	512
nilpotency class	3
isoclinism	isoclinic class of #23918
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^c=di,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, 109604);