#212917

order	512
nilpotency class	3
isoclinism	isoclinic class of #19733
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

722 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2,4,3)(7,13)(8,14)(10,16)(12,17)(15,19)(18,20)(21,22,26,30)(23,31,33,45)(24,28,35,42)(25,37,38,29)(27,39,41,34)(32,44,47,52)(36,49,48,43)(40,50,51,46);(5,6,9,11)(7,10,15,18)(8,12,14,17)(13,16,19,20)(21,23,25,34,26,33,38,39)(22,27,29,31,30,41,37,45)(24,32,36,46,35,47,48,50)(28,40,43,44,42,51,49,52);(5,7,14,19,9,15,8,13)(6,10,17,20,11,18,12,16)(21,24,26,35)(22,28,30,42)(23,32,33,47)(25,36,38,48)(27,40,41,51)(29,43,37,49)(31,44,45,52)(34,46,39,50);(1,3,4,2)(6,11)(10,18)(12,17)(16,20)(23,33)(24,35)(27,41)(28,42)(31,45)(34,39)(36,48)(43,49)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=fhi,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,d^c=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120113);