#128919

order	512
nilpotency class	4
isoclinism	isoclinic class of #11992
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1555
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8231
prime radical	#3147	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=i,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fgh,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214066, #72214124
maximal normal subgroup of	#72214066, #72214124
maximal quotient of	#72216475, #72216572

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36115);