#234459

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#120
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2553 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|d^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,e^b=eh,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231671, #72261762, #72261773
maximal normal subgroup of	#72231671, #72261762, #72261773

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141655);