#362876

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237227, #72239887, #72239894, …
maximal normal subgroup of	#72237227, #72239887, #72239894, …
maximal quotient of	#72237241

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270072);