#225389

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂²×D₄	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 132585);