#370025

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	#2580
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

772 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)(7,16)(8,20)(9,23)(10,15)(11,18)(12,19)(13,21)(14,24)(17,30)(22,34)(25,32)(26,33)(27,36)(28,35)(29,31);(1,3,4,2)(5,7)(6,11,15,26)(8,17)(9,18)(10,19)(12,14,16,23)(13,25,28,30)(20,29,34,27)(21,31)(22,32)(24,33)(35,36);(5,7)(6,12,15,16)(8,17)(9,18)(10,19)(11,23,26,14)(13,27,28,29)(20,25,34,30)(21,31)(22,32)(24,33)(35,36);(5,8,9,21,10,22,24,35)(6,13,14,20,15,28,23,34)(7,17,18,31,19,32,33,36)(11,25,16,29,26,30,12,27)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=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, 277221);