#185066

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
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×#3489	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

3570 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)(7,11,12,9)(8,10)(14,18,20,23)(19,25,26,28);(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,23)(17,20)(18,24)(21,26)(22,28)(25,27);(2,4)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(15,21)(19,26)(22,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239068
maximal normal subgroup of	#72239068
maximal quotient of	#72239145

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92262);