#396656

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1114 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)(5,6)(7,9)(8,10)(11,12)(13,14,17,23,19,15,18,21)(16,22,24,27,26,20,25,28);(5,7,8,11)(6,9,10,12)(13,15,19,14)(16,22,26,20)(17,23,18,21)(24,28,25,27);(6,10)(9,12)(13,16,18,25,19,26,17,24)(14,20,21,27,15,22,23,28);(2,4)(13,17,19,18)(14,21,15,23)(16,24,26,25)(20,27,22,28)

Polycyclic presentation:

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

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, 303852);