#403955

order	512
nilpotency class	3
isoclinism	isoclinic class of #393934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2874 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:

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

Polycyclic presentation:

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

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, 311151);