#407048

order	512
nilpotency class	3
isoclinism	isoclinic class of #407048
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14946
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1163 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 1163 conjugacy classes of subgroups — too many to draw. Only its 224 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 — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319453, #72319460, #72319471
maximal normal subgroup of	#72319453, #72319460, #72319471

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314244);