#238536

order	512
nilpotency class	3
isoclinism	isoclinic class of #232393
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	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=h,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=di,f^c=fh,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, 145732);