#370031

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	#2487
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

774 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)(5,6,10,15,11,16,9,13)(7,17,20,28,21,27,19,12)(8,22,24,30,25,29,23,14)(18,32,34,36,35,26,33,31);(1,3,4,2)(5,7,10,20,11,21,9,19)(6,12,13,27,16,28,15,17)(8,18,24,34,25,35,23,33)(14,26,29,36,30,32,22,31);(6,13,16,15)(12,17,28,27)(14,29,30,22)(26,31,32,36);(5,8,9,23,11,25,10,24)(6,14,15,22,16,30,13,29)(7,18,19,33,21,35,20,34)(12,26,17,31,28,32,27,36)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=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, 277227);