#93147

order	512
nilpotency class	4
isoclinism	isoclinic class of #93145
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂⁵	C₄²
Frattini, cosocle	C₂²×#124	C₂²
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁵ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂²×#124 ▷ C₂⁴ ▷ C₂² ▷ C₁

418 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2)(3,6,9,13)(4,11,12,7)(5,8)(10,14)(15,16)(17,18)(19,27)(20,31,32,24)(21,25)(22,26)(23,28,35,41)(29,39)(30,40)(33,44,45,37)(34,38)(36,42,46,48)(43,47);(1,3,4,10,5,9,12,15)(2,6,7,14,8,13,11,16)(17,19,21,29,22,30,34,43)(18,23,25,36,26,35,38,46)(20,28,33,42,32,41,45,48)(24,27,37,39,31,40,44,47)

Polycyclic presentation:

2^9|a^2=d,b^2=e,e^2=hi,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=gi,c^b=cg,d^b=df,f^b=fi,g^b=gi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268780
maximal normal subgroup of	#72268780
maximal quotient of	#72268783

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343);