#213238

order	512
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#127
prime radical	C₂³	#532
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

676 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 246 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fi,d^2=gi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210337, #72210421, #72210451
maximal normal subgroup of	#72210337, #72210421, #72210451
maximal quotient of	#72210308

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120434);