#6393443

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3549	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,e^b=ei,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257681, #72264169, #72264348
maximal normal subgroup of	#72257681, #72264169, #72264348
maximal quotient of	#72265240

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6300639);