#363917

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
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×#3492	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2782 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)(13,14)(15,21,22,28)(16,20)(17,19)(18,23,26,25)(24,27);(2,4)(5,6,8,9)(7,10,11,12)(16,24)(20,27)(21,28)(23,25);(6,9)(10,12)(13,15)(14,18)(16,25)(17,22)(19,26)(20,28)(21,27)(23,24);(2,4)(5,7,8,11)(6,10,9,12)(14,19)(16,24)(18,26)(23,25)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^b=fi,e^c=ei,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318071, #72318087, #72318100
maximal normal subgroup of	#72318071, #72318087, #72318100

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271113);