#372907

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

847 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 280103);