#245096

order	512
nilpotency class	3
isoclinism	isoclinic class of #245084
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1343 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1343 conjugacy classes of subgroups — too many to draw. Only its 176 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=dh,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227975, #72227979
maximal normal subgroup of	#72227975, #72227979

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152292);