#93196

order	512
nilpotency class	5
isoclinism	isoclinic class of #93192
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1316
commutator	C₂×C₄⋊C₄	C₂×C₈
Frattini, cosocle	#1955	C₂²
prime radical	#3496	C₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1955 ◁ G
LECS	G ▷ #1955 ▷ C₂×C₄² ▷ C₂³ ▷ C₂² ▷ C₁

292 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,4,9,6,7,13,14,5,10,12,16,3,8,11,15)(17,18)(19,26,27,31,28,32,29,22)(20,23)(21,25)(24,30);(2,7)(3,6)(4,12)(8,10)(9,14)(15,16)(17,19)(18,22)(20,29)(21,28)(23,26)(24,32)(25,31)(27,30)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392);