#94876

order	512
nilpotency class	2
isoclinism	isoclinic class of #94871
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

664 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ch,f^a=fg,c^b=ci,e^b=eg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#94876, #72211656
maximal normal subgroup of	C₂×#94876, #72211656
maximal quotient of	#72210932, C₂×#94876, #72283893

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2072);