#378490

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
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

903 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=efh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=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, 285686);