#185031

order	512
nilpotency class	3
isoclinism	isoclinic class of #160892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2292 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)(6,9,10,11)(13,14)(15,22)(16,19)(17,21)(18,26)(20,24)(23,27)(25,28);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,23)(17,24)(18,27)(19,25)(21,28)(22,26);(2,4)(13,16,19,14)(15,23,25,20)(17,21,18,26)(22,24,28,27);(13,17,19,18)(14,21,16,26)(15,24,25,27)(20,28,23,22)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239885
maximal normal subgroup of	#72239885
maximal quotient of	#72239898

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92227);