#475914

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214443, #72216163, #72216716
maximal normal subgroup of	#72214443, #72216163, #72216716
maximal quotient of	#72214149

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383110);