#242619

order	512
nilpotency class	3
isoclinism	isoclinic class of #233100
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149815);