#396655

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

1190 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)(13,14,18,19,17,22,16,21)(15,20,25,26,23,28,24,27);(5,7,8,11)(6,9,10,12)(14,19)(15,23)(16,18)(20,27)(21,22)(26,28);(6,10)(9,12)(13,15,16,24,17,23,18,25)(14,20,21,27,22,28,19,26);(2,4)

Polycyclic presentation:

2^9|a^2=ei,b^2=f,c^2=e,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.

Relationships

maximal subgroup of	#72322874
maximal normal subgroup of	#72322874

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303851);