#6372017

order	512
nilpotency class	3
isoclinism	isoclinic class of #60454
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#575	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255617, #72255619, #72255634, …
maximal normal subgroup of	#72255617, #72255619, #72255634, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279213);