#386851

order	512
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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

816 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:

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

Polycyclic presentation:

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

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, 294047);