#370003

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=eg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277199);