#128026

order	512
nilpotency class	4
isoclinism	isoclinic class of #7316
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1304
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
socle	C₂	C₂×#1304
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

1336 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1336 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,e^2=i,f^2=i,b^a=bd,d^a=dg,f^a=fh,h^a=hi,d^b=dh,e^b=eg,g^b=gi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72266652
maximal subgroup of	#72266596, #72266652, #72342770, …
maximal normal subgroup of	#72266596, #72266652, #72342770, …
maximal quotient of	C₄×#7317

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35222);