#504430

order	512
nilpotency class	4
isoclinism	isoclinic class of #504384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13341	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,3,4)(5,6)(7,11)(8,12)(9,14)(10,15)(13,22)(16,25)(17,26)(18,35)(19,28)(20,29)(21,31)(23,32)(24,33)(27,34)(30,36);(2,4)(5,7)(6,11)(8,16)(9,23)(10,17)(12,25)(13,24)(14,32)(15,26)(18,20)(19,21)(22,33)(27,30)(28,31)(29,35)(34,36);(7,16)(8,20)(9,21)(11,24)(12,29)(13,30)(14,31)(17,33)(18,19)(22,36)(23,34)(25,26)(27,28)(32,35);(2,4)(5,8,9,20,10,22,21,36)(6,12,14,29,15,13,31,30)(7,16,23,18,17,33,19,34)(11,25,32,35,26,24,28,27)

Polycyclic presentation:

2^9|a^2=e,d^2=fh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249933, #72252778, #72252788
maximal normal subgroup of	#72249933, #72252778, #72252788

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411626);