#238621

order	512
nilpotency class	3
isoclinism	isoclinic class of #233642
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

975 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,f^c=fh,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145817);