#93228

order	512
nilpotency class	5
isoclinism	isoclinic class of #93221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1343
commutator	C₂×C₄⋊C₈	C₂×C₄
Frattini, cosocle	#1955	C₂²
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂×C₄⋊C₈ ▷ C₂×C₄² ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1955 ◁ G
LECS	G ▷ #1955 ▷ C₂×C₄² ▷ C₂³ ▷ C₂² ▷ C₁

185 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 185 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2)(3,10,11,6)(4,8,14,7)(5,9)(12,16,13,15)(17,18,21,26,23,28,22,27)(19,29,31,45,33,46,32,42)(20,34,36,43,37,44,38,25)(24,35,39,30,41,47,40,48);(1,3,5,12)(2,6,9,15)(4,13,14,11)(7,10,8,16)(17,19)(18,24)(20,35)(21,32)(22,31)(23,33)(25,42)(26,40)(27,39)(28,41)(29,34)(30,36)(37,47)(38,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=d,b^2=h,c^2=fhi,d^2=hi,e^2=i,f^2=hi,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=cf,d^b=defi,e^b=ei,f^b=fhi,g^b=gi,d^c=dgi,e^c=ei,e^d=eh

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 424);