#94079

order	512
nilpotency class	5
isoclinism	isoclinic class of #7348
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1317
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂×#407	C₂²
socle	C₂	#7187
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ C₂×#407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

402 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,d^2=g,e^2=i,g^2=i,b^a=bc,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=dfi,g^b=gi,d^c=dh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1275);