#427303

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2002	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219642, #72219665, #72219666
maximal normal subgroup of	#72219642, #72219665, #72219666

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334499);