#475941

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

788 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=gi,c^2=f,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214451, #72216158, #72216718
maximal normal subgroup of	#72214451, #72216158, #72216718
maximal quotient of	#72214149

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383137);