#368785

order	512
nilpotency class	3
isoclinism	isoclinic class of #196350
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,11)(8,12,19,25)(9,13)(10,14)(15,21,28,32)(16,22)(17,23)(18,24,30,34)(20,26)(27,31,35,36)(29,33);(2,4)(5,7)(6,11)(8,15,19,28)(9,16)(10,17)(12,21,25,32)(13,22)(14,23)(18,27,30,35)(20,29)(24,31,34,36)(26,33);(5,8)(6,12,14,25)(7,15,17,28)(9,18)(10,19)(11,21)(13,24,26,34)(16,27,29,35)(20,30)(22,31)(23,32)(33,36);(2,4)(5,9,10,20)(6,13,14,26)(7,16,17,29)(8,18,19,30)(11,22,23,33)(12,24,25,34)(15,27,28,35)(21,31,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314372, #72314390, #72314414, …
maximal normal subgroup of	#72314372, #72314390, #72314414, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275981);