#94192

order	512
nilpotency class	5
isoclinism	isoclinic class of #94192
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1391
commutator	C₂²×C₈	C₄²
Frattini, cosocle	#2922	C₂²
socle	C₂	#7214
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ #2922 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

170 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgi,e^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gh,h^a=hi,d^b=dg,d^c=dh,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1388);