#93156

order	512
nilpotency class	4
isoclinism	isoclinic class of #93155
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	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#123	C₂²
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×#123 ▷ C₂⁴ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 449 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,e^2=h,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=gi,c^b=cg,d^b=dfh,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	#72268762
maximal normal subgroup of	#72268762
maximal quotient of	#72268765

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 352);