#93158

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×C₂²⋊Q₈	C₂²
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₂²⋊Q₈ ▷ C₂⁴ ▷ C₂² ▷ C₁

479 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^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	#72268763
maximal normal subgroup of	#72268763
maximal quotient of	#72268765

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 354);