#213043

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
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	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#2011	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

841 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=chi,d^b=dh,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319862, #72319873, #72320122
maximal normal subgroup of	#72319862, #72319873, #72320122
maximal quotient of	#72321162

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120239);