#238551

order	512
nilpotency class	3
isoclinism	isoclinic class of #232415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2656	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1066 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)(8,10)(11,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,27)(24,26)(25,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,24)(17,23)(18,25)(20,26)(21,27)(22,28);(1,3,4,2)(6,9)(7,12)(10,11)(14,20)(15,23)(16,17)(21,22)(24,25)(26,27);(6,10)(9,11)(13,16,18,17)(14,21,22,20)(15,23,25,24)(19,26,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145747);