#373641

order	512
nilpotency class	3
isoclinism	isoclinic class of #23897
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2200 conjugacy classes of subgroups; only the 346 normal subgroups are shown.

normal subgroups only

This group has 2200 conjugacy classes of subgroups — too many to draw. Only its 346 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) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 280837);