#361107

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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

886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=fi,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,g^b=gi,g^c=gi

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, 268303);