#513768

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁵×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4022 conjugacy classes of subgroups; only the 1602 normal subgroups are shown.

normal subgroups only

This group has 4022 conjugacy classes of subgroups — too many to draw. Only its 1602 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 — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72287301
maximal subgroup of	#72287301, #72287386, #72287856, …
maximal normal subgroup of	#72287301, #72287386, #72287856, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420964);