#454184

order	512
nilpotency class	3
isoclinism	isoclinic class of #454045
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

3473 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(6,9)(10,12)(17,20);(1,2)(3,4)(5,6,8,9)(7,10,11,12);(2,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20);(7,11)(10,12)(13,15)(14,17)(16,19)(18,20)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260718, #72317547, #72317550
maximal normal subgroup of	#72260718, #72317547, #72317550
maximal quotient of	#72260797, #72317723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361380);