#370004

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1022 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265390, #72307310, #72307350
maximal normal subgroup of	#72265390, #72307310, #72307350

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277200);