#92912

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	D₈
commutator	C₈	C₄×C₁₆
Frattini, cosocle	C₂×C₈²	C₂²
socle	C₂²	#1277
prime radical	C₂³	#334
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

170 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18)(19,28)(20,33)(21,36)(22,38)(23,27)(24,29)(25,35)(26,34)(30,40)(31,48)(32,44)(37,41)(39,47)(42,46)(43,45);(17,19,21,30,34,45,35,46,23,32,20,29,22,31,37,47)(18,24,25,28,38,42,36,48,27,40,41,44,26,39,33,43)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=fh,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225412, #72225511, #72225530
maximal normal subgroup of	#72225412, #72225511, #72225530
maximal quotient of	#72225409

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108);