#376656

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

843 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283852);