#356640

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

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×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,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, 263836);