#185099

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3506	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2634 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)(6,10)(7,8)(11,12)(13,14)(15,21,23,18)(16,19)(17,20)(22,27,28,25)(24,26);(5,6)(7,10)(8,12)(9,11)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(2,4)(5,7,9,8)(6,10,11,12)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28);(5,7,9,8)(6,10,11,12)(14,20)(16,24)(18,21)(22,28)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,g^b=ghi,d^c=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239068
maximal normal subgroup of	#72239068
maximal quotient of	#72239149

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92295);