#378075

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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 ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

918 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)(7,12)(8,10)(9,11)(13,14,20,24,18,23,19,17)(15,21,25,27,22,28,26,16);(1,3,4,2)(5,7,9,8)(6,10,11,12)(13,15,18,22)(14,21,23,28)(16,24,27,17)(19,26,20,25);(2,3)(5,8,9,7)(6,10,11,12)(13,16)(14,22)(15,23)(17,25)(18,27)(19,28)(20,21)(24,26);(6,11)(10,12)(13,17,19,23,18,24,20,14)(15,16,26,28,22,27,25,21)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=fh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,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, 285271);