#23123

order	256
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	Aut(G) = C₂¹³ (order 8192) · Inn(G) = C₂⁴ · Out(G) = C₂⁹
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁴
commutator, socle	C₂³	C₂³×C₄
prime radical	#392	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

478 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:

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

Polycyclic presentation:

2^8|a^2=g,b^2=g,c^2=ef,g^2=h,b^a=be,c^a=cf,c^b=ch,d^b=de,d^c=dh

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#23123, #6382556, #6382559, …
maximal normal subgroup of	C₂×#23123, #6382556, #6382559, …
maximal quotient of	#231158, #231159, #231160, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16111);