#242441

order	512
nilpotency class	3
isoclinism	isoclinic class of #233245
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1026 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=dh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149637);