#376397

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14,19,24)(15,25,29,34)(16,30,33,37)(17,20,18,23)(21,28,35,26)(22,32,38,31)(27,41,43,36)(39,42,44,40);(5,7,8,12)(6,9,10,11)(14,20)(15,26)(17,18)(21,35)(22,30)(23,24)(27,40)(28,29)(31,32)(36,41)(37,38)(42,43);(13,15,19,29)(14,21,24,35)(16,27,33,43)(17,26,18,28)(20,34,23,25)(22,36,38,41)(30,44,37,39)(31,42,32,40);(1,3,4,2)(6,10)(9,11)(13,16)(14,22)(15,27)(17,32)(18,31)(19,33)(20,30)(21,36)(23,37)(24,38)(25,39)(26,40)(28,42)(29,43)(34,44)(35,41)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283593);