#186030

order	512
nilpotency class	3
isoclinism	isoclinic class of #18252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3536	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309263, #72309266, #72309279, …
maximal normal subgroup of	#72309263, #72309266, #72309279, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93226);