#401007

order	512
nilpotency class	3
isoclinism	isoclinic class of #400057
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1860 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224753, #72224754, #72224756
maximal normal subgroup of	#72224753, #72224754, #72224756

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308203);