#403406

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2770 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)(15,19)(16,24,26,20)(17,21)(18,22)(23,27,25,28);(2,4)(5,6,8,10)(7,11,12,9)(13,15,18,17)(14,19,22,21)(16,25,26,23)(20,28,24,27);(5,7,8,12)(6,9,10,11)(13,16,18,26)(14,20,22,24)(15,23,17,25)(19,27,21,28);(13,17,18,15)(14,21,22,19)(16,25,26,23)(20,28,24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268102, #72317885, #72317892, …
maximal normal subgroup of	#72268102, #72317885, #72317892, …
maximal quotient of	#72319655

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310602);