#405777

order	512
nilpotency class	3
isoclinism	isoclinic class of #393929
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2200 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)(7,9)(8,10)(11,12)(13,14)(15,22,19,16)(17,20)(18,21)(23,28,26,24)(25,27);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,16)(17,23)(18,19)(20,24)(21,22)(25,26)(27,28);(2,4)(13,16)(14,19)(15,21)(17,24)(18,22)(20,26)(23,27)(25,28);(6,10)(9,12)(13,17,18,25)(14,20,21,27)(15,23,19,26)(16,24,22,28)

Polycyclic presentation:

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

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, 312973);