#355612

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=efg,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di,f^c=fi,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, 262808);