#356840

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1152 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,f^c=fi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264036);