#110576

order	512
nilpotency class	3
isoclinism	isoclinic class of #9383
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1154 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=gh,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238945, #72258125, #72258127
maximal normal subgroup of	#72238945, #72258125, #72258127
maximal quotient of	#72238980

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17772);