#371422

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1149 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)(13,14,20,23,19,17,18,27)(15,24,25,21,22,26,28,16);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,21)(16,17)(18,28)(19,22)(20,25)(23,26)(24,27);(6,9)(10,12)(13,16,19,21)(14,22,17,15)(18,24,20,26)(23,25,27,28);(5,7,8,11)(6,10,9,12)(13,17,19,14)(15,21,22,16)(18,23,20,27)(24,25,26,28)

Polycyclic presentation:

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

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, 278618);