#368791

order	512
nilpotency class	3
isoclinism	isoclinic class of #196143
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3055	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1800 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238735
maximal normal subgroup of	#72238735
maximal quotient of	#72238750

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275987);