#374060

order	512
nilpotency class	3
isoclinism	isoclinic class of #196332
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35250	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2632 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260446, #72260459, #72262285, …
maximal normal subgroup of	#72260446, #72260459, #72262285, …
maximal quotient of	#72260683, #72262014

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281256);