#371439

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1135 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)(13,14,17,21,18,16,23,20)(15,22,26,24,27,28,25,19);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,23,25,18,27,17,26)(14,19,20,28,16,24,21,22);(6,9)(10,12)(13,16)(14,18)(15,24)(17,20)(19,27)(21,23)(22,25)(26,28);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,18)(15,24)(17,20)(19,27)(21,23)(22,25)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278635);