#402197

order	512
nilpotency class	3
isoclinism	isoclinic class of #397902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂³×D₄	Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1326 conjugacy classes of subgroups; only the 248 normal subgroups are shown.

normal subgroups only

This group has 1326 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238735
maximal normal subgroup of	#72238735
maximal quotient of	#72238751

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309393);