#381697

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

775 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)(6,12,15,24)(13,16,29,26)(14,27,31,32)(28,30,36,35);(5,6,9,24,11,15,10,12)(7,16,18,13,20,26,19,29)(8,14,22,32,23,31,25,27)(17,30,21,28,33,35,34,36);(5,7,11,20)(6,13,15,29)(8,21,23,34)(9,19,10,18)(12,26,24,16)(14,30,31,35)(17,25,33,22)(27,28,32,36);(1,3,4,2)(5,8,9,22,11,23,10,25)(6,14,24,32,15,31,12,27)(7,17,18,21,20,33,19,34)(13,28,26,35,29,36,16,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288893);