#380782

order	512
nilpotency class	3
isoclinism	isoclinic class of #29742
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,23,19,24,17,22)(15,20,28,36,26,34,27,35)(16,21,30,38,31,39,32,40)(25,33,29,37,41,43,42,44);(6,9)(10,12)(13,15,17,27,19,26,18,28)(14,20,22,35,24,34,23,36)(16,29,30,41,31,42,32,25)(21,37,38,43,39,44,40,33);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,21)(15,25)(17,32)(18,30)(19,31)(20,33)(22,40)(23,38)(24,39)(26,41)(27,42)(28,29)(34,43)(35,44)(36,37);(7,11)(10,12)(15,26)(20,34)(25,41)(27,28)(29,42)(33,43)(35,36)(37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214470, #72216185, #72216736
maximal normal subgroup of	#72214470, #72216185, #72216736
maximal quotient of	#72214159

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287978);